Problem: What is the remainder when $2001 \cdot 2002 \cdot 2003 \cdot 2004 \cdot 2005$ is divided by 19?
Reducing each factor modulo 19 first, we see that $2001 \cdot 2002 \cdot 2003 \cdot 2004 \cdot 2005 \equiv 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10 \equiv 30240 \equiv \boxed{11} \pmod{19}$.